Maybe just twist a 222 faster? The animals certainly won't notice the 222 case.
The reality is that going from a 222 to about anything it spawned is "slight", and doesn't get appreciable until you get to 250 sized hull capacity. If you really want more power/speed, all else being equal, going from a 222 to a 223 is closer to nothing (100fps at like pressure) than when going from 223 to the 250 case in ANY form. Those are just hard mathematical realities.
If decent 222 cases are that hard to find (which they aren't), 5.56 brass is easily snugged and trimmed into 222 cases.

The 22 Nosler sucks with tipped bullets (just like the 223/223AI) in an AR, but would be a little quicker than anything 222 based in a 700 and would not require changing your bolt. I would throat it beyond SAAMI specs for intended bullets and change the mag over to the 308 configuration. You'd obviously be stuck with one case maker.

Beyond that, the 250 hull is your best bet for any real gain.